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A258279 Expansion of psi(q)^2 * chi(-q^3)^2 in powers of q where psi(), chi() are Ramanujan theta functions.

%I A258279 #11 Feb 16 2025 08:33:25
%S A258279 1,2,1,0,-2,-2,0,0,1,-4,-4,0,0,4,0,0,-2,-2,4,0,2,0,0,0,0,6,2,0,0,-2,0,
%T A258279 0,1,0,-4,0,4,4,0,0,-4,-2,0,0,0,-8,0,0,0,2,3,0,-4,-2,0,0,0,0,-4,0,0,4,
%U A258279 0,0,-2,-4,0,0,2,0,0,0,4,4,2,0,0,0,0,0,2
%N A258279 Expansion of psi(q)^2 * chi(-q^3)^2 in powers of q where psi(), chi() are Ramanujan theta functions.
%C A258279 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A258279 G. C. Greubel, <a href="/A258279/b258279.txt">Table of n, a(n) for n = 0..1000</a>
%H A258279 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258279 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258279 Expansion of eta(q^2)^4 * eta(q^3)^2 / (eta(q)^2 * eta(q^6)^2) in powers of q.
%F A258279 Euler transform of period 6 sequence [ 2, -2, 0, -2, 2, -2, ...].
%F A258279 G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 36 (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A002175.
%F A258279 G.f.: Product_{k>0} (1 - x^(2*k))^2 / (1 - x^k + x^(2*k))^2.
%F A258279 Convolution square of A089810.
%F A258279 a(2*n) = A258228(n). a(3*n + 1) = 2 * A258277(n). a(3*n + 2) = A258278(n). a(4*n + 3) = 0. a(6*n + 2) = A122865(n). a(6*n + 4) = -2 * A122856(n). a(12*n + 1) = 2 * A002175(n). a(12*n + 5) = -2 * A121444(n).
%F A258279 a(18*n) = A004018(n). a(18*n + 3) = a(18*n + 6) = a(18*n + 12) = 0.
%e A258279 G.f. = 1 + 2*q + q^2 - 2*q^4 - 2*q^5 + q^8 - 4*q^9 - 4*q^10 + 4*q^13 + ...
%t A258279 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/6, q]^2, {q, 0, n}];
%o A258279 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)))^2, n))};
%Y A258279 Cf. A002175, A004018, A089810, A121444, A258277, A258278.
%K A258279 sign
%O A258279 0,2
%A A258279 _Michael Somos_, May 25 2015